Sufficient conditions for the existence of solutions for a thermoelectrochemical problem

Abstract

Abstract. A mathematical model is introduced for thermoelectrochemical phenomena in an electrolysis cell, and its qualitative analysis is focused on existence of solutions. The model consists of a system of nonlinear parabolic PDEs in conservation form expressing conservation of energy, mass and charge. On the other hand, an integral form of Newton's law is used to describe heat exchange at the electrolyte/electrode interface, a nonlinear radiation condition is enforced on the heat flux at the wall and a nonlinear boundary condition is considered for the electrochemical flux in order to account for Butler-Volmer kinetics. The main objective is the nonconstant character of each parameter, that is, the coefficients are assumed to be dependent on the spatial variable and the temperature. Making recourse of known estimates of solutions for some auxiliary elliptic and parabolic problems, which are explicitly determined by the Gehring-Giaquinta-Modica theory, we find sufficient smallness conditions on the data to guarantee the existence of the original solutions via the Schauder fixed point argument. These conditions may provide useful informations for numerical as well as real applications. We conclude with an example of application, namely the electrolysis of molten sodium chloride.

Weak solutions for multiquasilinear elliptic–parabolic systems: application to thermoelectrochemical problems

“…for every i, j = 1, · · · , I + 1, and for all v ∈ H 1 (Ω). Thanks to Propositions 5.1, 5.2 and 5.4, we may pass to the limit in (47) and (49), as M tends to infinity, concluding that u i and φ verify, respectively, (7), for i = 1, · · · , I, and (9).…”

“…Here, h C denotes the conductive heat transfer coefficient, θ l denotes a prescribed surface temperature, g i,l may represent a truncated version of the Butler-Volmer expression (cf. [7,8] and the references therein), and g denotes a prescribed surface electric current assumed to be tangent to the surface for all t > 0. The parabolic-elliptic system (56)-(58) is accomplished by (59) and the remaining boundary conditions.…”

“…The mathematical modeling of electrochemical devices such as Lithium-ion battery system [15,22] has gaining of interest in the literature [26,27,33]. Here, no internal interfaces are considered in the model, which amounts to neglecting possible material heterogeneities as done in [6,7,8]. These works deal with weak solutions related to thermoelectrochemical devices with radiative effects in a part of the boundary, involving the cross effects.…”

“…A particular feature is the mixture of some kind of (nonlinear) Neumann and Robin boundary conditions. Also, quantitative estimates are stated for the norm (steady-state in [6] and unsteady-state in [7]) under appropriate assumptions on the data, where the constants are given explicitly. Within this state of mind, we close this paper by applying the theoretical coupled elliptic-parabolic system to the thermoelectrochemical phenomena.…”